the precision measurement of transformer ratios

1960 Cutkosky and Shields: The Precision Measurement of Transformer Ratios 245

lo (SOR 9 (Sb CO QUIV T CONTC lST C LIO

rC = /\ SJ | r II HS

rO=40S(+ a ~) Sb R-0-(IS)(______ 171/

so 'R(IIQ- Q 0 2 3

*(0 *(S b- *) EQUIVALENT CONTACT RESISTANCE IN MILLIOHMSa. g23c3otiuino cnatrssac

so~~~~~~~~~~~~~~~~~~vrain to400ait deitin

I )~ II EQUIVALENT CONTC RESISTANC EIN ARIATIONM

H- ±0.027 PPM_______ 400~SbAm TO THE 3o

ZOOz A0s | r 400_-s) >nte posbesuc ftobei otc eit

Amarb var -ITS400---1400 LIIT

4-_r _I___I- _ _

0 2 3

Fig. 21-Effect of one resistor in a ten-step voltage divider. EQUIVALENT CONTACT RESISTANCE IN MILLIOHMS

Fig. 23-Contribution of contact resistance

n~~~~~0 400

whatprec variationse tolnesariy deition.eifrprtpr

ance variation. To minimize this problem all switchcontacts of the Model RV-622 are doubled. Measure-

The PrecIsIoMeasurem ments of large numbers of contacts and a statisticalI. INTRODUCTION kanalysis of the results have revealed the results shown

in Fig. 23. A linearity deviation of less than 0.025 ppmthevalueof the first is by means ofaprecise,stabl can be expected from contact resistance variations on a

series impedances.. Scam sreteue 10 kilohm ESI Model RV-622 Six-Decade VoltageDivider.

z ~~~~~~~~~~~~~~VIII. CONCLUSION

SETTING S--t Now the voltage divider is calibrated, and we know

Fig.l22-How the linearity changes when the what precautions are necessary to use it for part-per-

Fig.nSadad n lcrni esrmnsasppr54 hmiloentme pasurmets.o h 1g otg wnl h

following decade is switched. millismeancemena r d

nThePrecision Measurement of Trans ormer atiosR. D. CUTKOSKYt AND J. Q. SHIELDSt

I. INTRODUCTION knowledge of the open-circuit voltage ratio and equiva-A rCONVENIENT and accurate method for cali- lent series impedances of the transformer.

brating a given capacitor in terms of another ca- One method for measuring a nominally ten-to-onepENTAcItN,vor. havIngap. val; tenimbes o. oe t transformer ratio has been described by Thompson.13

th paluetof theafirst is bylumeans ofmea precise, stable This method consists of determining the relative valuestransformer with a ten-to-one ratio and low equivalent o lvnnmnlyeultretria aaiosbseries impedances.1' Such a measurement requires a simple substitution and then setting up a bridge with

244 IRE TRANSACTIONS ON INSTRUMENTATION September

DET

Fig. 1-Basic transformer bridge circuit. n X fn- I

E,tions are applied for the effect of finite lead impedances Fig. 2-Equivalent circuit of measuring systemanid load admittances, yields a value for the open-circuit employing permutation method with s = 1.transformer ratio.

If we are to obtain reliable ratio measurements, onerequirement is that all of the capacitances involved switch position. Balancing the bridge with a small ad-drift at nearly the same rate. This can be achieved by mittance ys, assumed for simplicity to be connected toconstructing them to be identical and mounting them the transformer winding of voltage E1, we havein good thermal contact with each other.

It can be shown that if eleven nominally equal three- is= -Elysterminal capacitors are mounted in this fashion withtheir detector leads connected together, and if elevenbalances are made with the line lead of each capacitor m+n m+nin turn on the high-voltage side of a 10:1 transformer E1 Z ys + (mEl + nE2) ZJYk = 0.

s=1 k=land the other ten line leads on the low-voltage side,one may obtain directly the voltage ratio at the points Making use of the relationshipof interconnection independent of mutual coupling andground impedances within the capacitor housing. The m(1 + a + j0)jfollowing section will show in detail how this is accom- E+ nplished in the more general case of a transformer with anominally m:n ratio. The technique is similar to thatused to establish known dc resistance ratios. m+n

ZY8II. THEORY OF PERMUTTATION METHOD a + jV = - - (1)

mn MYkFig. 2 shows a transformer with open-circuit ratio m £ Ykk=1

E2 mk=l

- --M _--- (1 + a + jO), where bars denote averages over the m+n indices.El ~~n

where m and n are integers and a and d are to be deter- III. LEAD IMPEDANCE CORRECTIONSmined. Connecting m+n admittances Y1, Y2,

I

Yk, Ym+n as shown, we have with the detector In the derivation of (1) in the previous section, it wasvoltage zero, assumed that the voltage ratio at the points of inter-

connection is constant and independent of the permut-n=m,nnr ing switch position. Although this requirement can be

k=n+l k=l satisfied, it is not necessary for the accuracy desired.Formulas will now be developed which relate the openIf the admittances are permuted cyclically, each ad- circuit transformer parameters to the average balanc-

mittance Yk, is replaced by Yk±+, and Ym±l+n is ieplaced ing admittance and the average loading associated withby Yi. If this permutation is repeated m+-n tinmes, each the assembly of permutable capacitors called the "ratioadmittance appears with voltage E2 n timles, anld with device" in the sequel.voltage E1 m times. We then have, after adding the Lttevlae tteitroncinpitm±n equations, after permuting s times be given by E(1( +e8') andm±n m±n mi-n mi-n E2(1+e8"'), where E1 and 132 are the open-circuit volt-£ is = mEi Z Yk + nEl £I Yk = (mE1 + nE2) £ Yk, ages and e8' and e8" are functions of the lead impedancess=l k=l k=li k=l and load admittances. Then the short-circuit detector

where is is the short-circuit detector current for the sth current corresponding to this configuration is given for


the special case m= 1O and n=1I by El2Z E,( I +e's)18 El(1 -1-es') 5i: Yk + E2(1 ± e,") Y,

k#s

E,E(I + e,1) (ZY,. - Y, Y

0El(t + a ± j3)(I + es")Y, Ywlwhere k runs from one to eleven. Balancing with an ad-mittance Ys assumed to have a voltage El, we have, 7

after summing over all s from one to eleven, S

Z Ys -I Z (1+ Z,) Yk - y- 1OZE(1 + e1") Y8 ______s k SE z E(I +e's)

- tO(a + j6) (1 + e,1') Y. = 0.s ~~~~~~~Fig.3-Equivalent circuit of measuring system showing

Writing Y,= Y, +AYs, we have lead impedances and load admittances.

ZYs -I 10 (e8' - e/') Z:Y -jZ AY,(l0es1' ± es1)

k 8 y

where=a- j[ s'+ (Al + 1)" + 'El

lbsYs~~~~~~~~~~~~~~~~~~~~~~

where eZ Y8e8[' __ __( 1+14Ysf/

t AY,( 8" b +e8 8T 2[i's + (i +I) F8"']1 0.

57, ~~- ZAYs(lOeffe-Ft+110 10s Y'],zir"~vl( 8

Writing y=jwc8+g8and Y5=jC5+G5, ad makingZ1Z2(ys5'ys + ys'ys"' + yst"ys/")theacitassuptio thaGe, +OeSae_______________

+ e,1 e,11~ ~ ~ ~ ~ ~ ~~~1+ $

Ys (I+ )_10Yee+E+E, whr10C8 b0wC~~~~~~~ =e Z2(ys5' + Ifs' + Z2ys" F1'

jG AYSThscioCs bY5 2 sa+jt3 +Z[y"+1 Y

Writing ys=jwcs+g, an Y,=jwC,+G, acd makinwinintrctansfmtoreandthet ratioC deviehatvhe lth z1[Ys5+lys"' +s E + E2i+ E3,)

switcpostion E1adE=g M1 1(1aj3E wherearehe oen-ircui voltageOs;Z1and Z2Yare theZimped -1 /s_lf

are 3ereetthelaaditnesqofvthenrti deirce.t Af motreac -8\1 +YS/+11Y"]+El+12+I3linedoncing thtransformertodthe ratio device,ah buthLF"+ +M -z1F'+( 1

points. Solvtiong frEl'and e2=-M,=whave aS\ Mlher


Writing __

1Y8' = jOcL ±Gs Eb j Eb

Y = jwC' " + GS/ I

YV"' = jCsC"' + Gsf' 7b Gb

Zi = jwLl + R, Y=jwcZ2 = jwL2 + R2,

we have, after separating real and imaginary parts, G a

=a cs+ W2Li(C,' + 11C,"') - C2L2(Cs" + 1.1C8"') E a alocs

-R1(Gs' + IlG8"') + R2(Gs" + 1.lGs"') + Re (eo), (2)Fig. 4-Equivalent circuit of transformer showing

g8 __ impedances and interwinding capacitance.=- ___ -cwR,(C,' + 1iC8"') + coR2(Cs," + 11Cs,"')

lOwCs secondaries having the turns of one side in close prox-- coLi(G8' + llG8"'t) + xL2(G" + l.1Gs8") imity to the turns of the other in order to reduce leak-

Im (EO0), (3) age inductance.2 This construction produces a trans-+ Im (eo), (3) former with a large distributed capacitance between

where windings.l0= el + E2 + 3. Fig. 4 represents the two secondary windings of a

transformer of voltages Ea and Eb. C represents the in-Making the assumptions that terwinding capacitance and Ey is the open-circuit volt-

aX + j# < 1 X 10-4 age between the two leads Ga and Gb. For use in a bridgecircuit, the terminals Ga and Gb of the transformer are

Zl Z21 < 0.1 ohm connected together and to the ground. We then have'Y|, Y| , Y"' < 1 X 10-3 mhos with

9X 10-8 mhos <K Y, < 11 X 10-8 mhos YZa7 and YZbI <<1,'A V.

<l 10XJ- 4 Eb' Eb+EYZbY EbF Ey EyEa' Ea + EyZaY Ea[1+ Eb EZa

G, ~~~~~~Eb[ (Ey 7J-Ey a]tG < I X 10-9, I-[ + jWJC(- Zb -Ea )coc, Ea ~ Eb Ea

we can show that Ey is typically quite large and for a 10: 1 transformer

< 4.3 X 10-5 Y' max + 8.1 X 10-1 has a value between Ea and Eb. C is unavoidably largebecause of the close spacing between windings, and may

|Y" max + 7.3 X 10-4 be as great as 0.02 ,f. From the above comments it can

IY"' max + 120 |YS , (4) be seen that the stability of the transformer ratio de-pends upon the stability of Za and Zb.

where IY'| max is the largest of the eleven values for For this reason, it is necessary for Za and Zb to beI Y,'f, and Y"| max and Y"'| max are defined simi- small and well defined. This can be accomplished bylarly. It should be observed that Eo is the error caused connecting the two secondary windings together per-by neglecting certain terms in mathematical expan- manently within the transformer housing with shortsions, and does not represent an inherent limitation of low-resistance leads.this method. With the above connection made and a single lead

In general, it is desirable for the correction terms in brought out from this connection, the complete trans-(2) and (3) to be small. The impedances may be signifi- former with its primary can be represented by thecantly reduced by careful attention to the methods of equivalent circuit based on an ideal transformer asconnection. For this reason, it is necessary to digress on shown in Fig. 5(a). A, B, and C represent the terminalsthe subject of transformer design. of a connection block to which the secondary leads are

attached. The circuit of Fig. 5(a) is equivalent to thatIV. TRANSFORMER DESIGN of Fig. 5(b). The accompanying formulas show that the

Precision transformers with low series impedances common impedance i; enters into the series impedanceare commonly constructed with heavy strap-copper terms, but not in a way basically different from t1 and


P V~~~~272P2 ~~~~~~B/A A

/

a _4B EBj / / /* B10 °N 1N- \i G L |G

P3~~ EA |t DA

(a) ~~~~~~~~~~~~~~~~~~(a)

I *B - B

= ~~~~~~~~~~~~~~~~~~~E B |

z P G

A

AAAA~ ~ ~ ~~E

l'~~~~~~~~~~~~~~~~~~~~~~' A_Ak . }1~~~~~~~~~~~~~~~~~~~~~~+ X33

(b)(b) Fig. 6-Use of high-permeability core to

r1=¢1+ 4 1+) 2'= +¢P(i P ) reduce equivalent impedances.P2 Pi

¢3 - ¢4 (¢3 pis -p33) ing one of these leads with one of the high-voltage leads

z G~~

¢3/= ---- - by means of a high-permeability core. This is electri-1- p3 ¢ cally equivalent to breaking the connection at p. Fig.z ~~~~~~~~~6(b)shows the resulting equivalent circuit of the trans-

(pl')3 = pi3 1-p3 4) (P2')3 = P23 P3- 3t- former whose terminals are At B and G of the connec-z z ~~~~~~~~tionblock. The terms Zl+ Z3 and Z2+ Z4 are the equiva-(p3'1)3 = p33 2_1_ (Z')3 = Z3 (1 _p3 _ lent series impedances of the transformer, and EBIEA is

I1 - p3 -4 the open-circuit voltage ratio we desire to measure.,, Since the value of EBIEA depends upon the loading ad-

P1P2P3 = PItP2 p3 = 1. ~~mittances present in the transformer, and its leads toFig. 5-Ideal transformer representation for three-winding trans- tecneto lc,teeamtacsmsewlformers, where the p's are ideal transformer ratios. tecneto lc,teeamtacsms ewl

defined. This is generally accomplished by shielding all

~2- ithrefrene toFig 5() ad th asocitedparts of the transformer and maintaining a fixed ge-

formuls, theparameers ofinteret areometry by means of rigid construction.formulas thepaametersof inteestareIt can be seen from the pictorial nature of Fig. 6(a)

p2' P2 that the equivalent series inductances of the transformer

A~~~t = 102t1t vl 1.14, ad t2 ~ 2+ 114- EaIerdcdb euigtesae ra ntefg

small as possibleThis may be acheved in effeFig. 6-Use ofeacin high-ptermeabilitycore to emes

bringin seart(round ingdonem ofe thesen leads withgonsoyhehihvotaeledneto -on bymehihetansfomrhuigt h h cofxa ladpuhigh-permeabilitycore.Thsiseectr-

connec ~ ~~ /tion block. h termsna Z,asin+Z.6() andlink-aiua lob rftbyue oredthe equeciva-


impedances from the connection block to the ratio de- gs Cs'_\vice. The use of high-permeability cores in a related ap- += 1(R -+ I.1C,"'plication has been discussed by Thompson.1

( Cs (Gs'V. LEAD IMPEDANCE AND LOAD + coR2 CS --_ +c ( + l. t GsADMITTANCE MEASUREMENT 10 10

Having discussed methods of obtaining small, well- / , s'defined impedances, we next turn our attention to the + wL2 tGs" -- + Im (EO). (9)measurement of these impedances. One method ofmeasurement may be developed by reference to (2) of Since the loading admittances associated with theSection III. If Cs"', for example, is increased by the leads can be made very small, the quantities C8" - C//tOaddition of components within the ratio device (beyond and G,"-G,'/10 in (8) and (9) will also be very smallthe permuting switch), one obtains and, consequently, the accuracy to which L2 and R2

need be measured is greatly reduced.=c~+AC8± w2LdC8' The principal significance of this procedure for the

ae = 10C + c 2L,(C,f + lilCs"' + 11 AC,8") measurement of 2 and (R is that any component added___s past the interconnection points produces essentially no

- 2L2(Cs" +-1.1Cs8" + 1.1ACS//) error by virtue of its mutual inductance with other por-

R1(Gs' + 11Gs"'') + R2(Gs'l + 1.1Gs8M) tions of the circuit. Mutual inductance between the-Re (co + AEo), added component and other elements of the ratio device

+ Re (EO + AEO), (5) changes the effective values of the transfer admittances

where AC,"', Ac8, and A\eo are the increases in C8",, c Yk defined in Section II, but the derivation is still validand Eo, respectively. Subtracting (2) from (5) and rear- if these new values for the Yk are used. In practice,ranging terms, we obtain since Yk is needed only to an accuracy of a few per cent,ranging_terms,_we_obtain the effect is insignificant.

Acs Re (IAEo) Errors resulting from mutual inductances between- 0L 2C1=(AC,"') +. _2(_AC_ (6) the added component and the leads preceding the in-

Similarl1,1fr(mC("') terconnection points also cancel to first order, since anySimilarly, from (3), change in the voltage at the interconnection points,

Ag8 Im (,AE0) caused by such mutual inductance, reverses sign whenR2- 10R1 = 2 ,,,-- - ,,, * (7) the component is switched from one side of the trans-

lco2CS (ACS"') 1. lw(IC8") former to the other. It can be seen that the net effect is

One might suppose that if similar measurements to leave the average voltage ratio at the interconnec-were made with components added beyond the permut- tion points independent of such mutual inductance.ing switch to increase C8", for example, the value of L2 It can now be seen that the only mutual inductancesby itself might be obtained with comparable precision which may produce errors in the measurement of a andHowever, since the permutation process places each d are those between the nonpermuted portions of theadded ground capacitance on the high-voltage side circuit. Since the nonpermuted circuit consists almostonce and on the low-voltage side ten times, a change of entirely of coaxial leads, we may expect all errors causedAC," in C8" produces a corresponding change of 10 by mutual inductances to be negligible.ACs' in C.', so that one arrives again at values for Since it is difficult to reduce the load admittances ofL2-10L1 and R2-10R1. In fact, were it not for the the leads and switch to zero, values of L2 and R2 areloading admittances associated with the wiring be- still required in formulas (8) and (9), but relatively lowtween the permuting switch and the connection block accuracy is sufficient. One method of measurement maywe would have C8'-=10C," and G,'-10G,", in which be developed by referring to (8). If components arecase knowledge of L1, L2, R1, or R2 alone would not be added outside the ratio device so as to increase C " byneeded. This fact can best be seen if we let 2 _L2 -10L1 an amount AC," wTith corresponding increases Ac8 andand cRsR2-10RI, and rewrite (2) and (3) in the fol- Aeo, one obtainslowing manner:cs_ __,22(-+ Ac84A+ _ /C-'

lCs 10 s.C,__~i 101_C

-cw2Li(Cs/ G') +(R/(1+lGs/) _ w2L2 (C8"/ + AC8" -__

+R2(Gs/--)+ Re (eo), (8) + Re(Eo+± GAEo). (10)G/G


Subtracting (8) from (10) and rearranging terms, we VI. NUMERICAL RESULTSobtain The procedures developed in the preceding sections

-c8 Re(A~i) of this paper have been applied to the measurement ofAC, R,(,AEO)L2= --_ + _ . (11) a particular transformer, NBS 89785. This transformer

10W2C8(AC/") W2(AC"//) was built according to the principles described at the

Similarly, from (9), we have 1958 Conference on Electrical Aleasurements2 exceptthat the thickness of the copper-strap secondary wind-

Ag, Im (A (1o) ing has been reduced to 0.030 inch to allow more roomR2 = - - ,, (12) for insulation.

t0o2C,AC,"1) w(,ACI) The ratio device and the transformer were placed in* , // d an o~~~~~ilbath with +0.01°C regulation. This decreasedThe capacitance coefficients C' and C" in (2) and an ol b

(3) represent principally the ground capacitances of the the drift rate of the capacitors in the ratio device, andt epes and a

* b t lo

hence improved the accuracy attainable with the meas-ratio dlevice and are determinedt by the design Of thecomponent capaciors.Cpacitoscoulbedeig uring system. Since the transformer ratio is slightlycomponent capacitors. Capacitors could be designed

with ground capacitances significantly smaller than temperature-sensitive, temperature control of the trans-

those incorporated in the ratio device, but this would former was also required (see Fig. 8).probably be at the expense of their stability. C"' could The work described below was all done at an angularbe made zero by careful shielding, but it is normally frequenicy of 104 radians per second (about 1592 cps).very small and causes no particular trouble in measure- The voltage level was monitored with an expanded-ment. We are faced then with the problem of measur- scale vacuum-tube voltmeter accurate to ± 4 per cent.ing C,', C,", and cv". The small variable admittances, c, and gs, consistedIn use, the ratio device is operated with the bridge of the low-range decade capacitor and the conductance

balanced, meaning that the detector voltage is zero. balance control described by McGregor, et al.,2 but al-Thus, a correct set of values for C', C", and es"' may most any capacitor variable in steps of 10-6 pf and ac-bo s s curate to 10t- pf would have been sufficient. The samebe obtained if the detector terminal iS shorted toground. The ratio device then has three remaining applies to the conductance balance control, which must

grounds .4, have steps of 10-8 ymhos accurate to 10-9 umhos. Sincetermina!sto whi are attached the lease from Ab only a small total range of the variable admittances isand G of the connection block. Representing this b)y . ..Fig. 7, we see that C,"' is the easiest coefficient to meas- required, this is not difficult to achieve.ure and is just the direct capacitance between points 4 In Section II, it was assumed that the small variableand B. C,"' is the average of the eleven measurements of admittances were coninected to the low-voltage side ofC."' corresponding to the eleven switch positions. C,' the transformer. It is necessary to reverse the signs ofcan be determined in practice by shorting B to G and these admittances in order to balance the bridge at

measuring C,'+C,"'. This can be done with simple two- certain switch positions. \We can accomplish this byterminal capacitance measuring techniques. Since the connecting one or both of the variable admittances toterminal

*caaiac.esrn ehius ic hthe high-voltage side of the transformer, but thistransformer ratio is defined at the connection block, t

the desired value for C,'+ C,"' is the increase in grouncd changes the apparent scale of the variable by a factorcapacitance of the connection block when the ratio de- of ten as well as changing the sign. The transformervice is connected to it. under investigation had a tap on the high-voltage side

Averaging the eleven measurements, we obtain C,'+ of the transformer at which the voltage was approxi-Cs "',from which C,' can be determined, and similarly, mately equal to that of the low-voltage side, but 1800C,". C8', G", and C,"' may be obtained simnultaneously out of phase. Use of this tap for the balancing admit-from the same type of measurement. tances, when required, greatly simplified the reduction

of data.Since the variable admittances have, in addition to

their direct admittances, relatively high admittancesto ground, it is necessary to measure these ground ad-

A - - B mittanices as well as the effect on the ten-to-one trans-,ll ~~~~~~former ratio of a ground admittance at each of the trans-Cs ~~~~~former terminals to which the variable admittances

_ s s-_~~~~~~maybe connected, and apply appropriate corrections.If it is borne in mind that the average voltage ratio isthe quantity of interest (see Section III), the correctionprocedure is straightforward. It involves, in additionto the quantities mentioned above, a knowledge of how

Z, ~~~~~manytimes each of the variable admittances is on eachG low-voltage transformer tap.

Fig. 7-Ratio device admittances with detector terminal shorted. The following parameters were determined for the

250 IRE TRANSACTIONS ON INSTRUMENTATION Septemberdescribed system:

2 = - 3.49)X 10-6 henry CR = - 66.3 X 10-s ohms

L2 = 1.0 X 10 5 henry R2 = 50 X 10-3 ohms

= 1101 X 10-i2 farads G/' = 44 X 10-9 mhos

C' = 127.1 X 10-12farads G-" = 5 X 10-9 mhos

C!" = 18.3 X 10-12 farads G"' = 1 X 10-9mhos.

Correction to a for loading of variable admittances--4X 10-9.Correction to : for loading of variable admittances- 19X 10-9.Total correction to a = +39X 10-9. Fig. 8-Photograph of measuring system. From left to right in oil

Total correction to d= -97 X 10-. bath: ratio device, connection block, and transformer.

These corrections were determined several times at Attempts to repeat the numbers presented in Tablesvarious temperatures and voltages. They were ob- I and II were not completely satisfactory, the discrep-served to be constant to ±1X10, over the range of ancies occasionally being as large as 0Xo10-9. Thor-conditions reported below. ough investigationi of the measuremenit equipment dis-

c, and g, were determinied from twenty-one separate closed no sources of error large enough to account forbridge balanices, with the ratio device switch running discrepancies larger thani 2 X 10. It is believed thatfrom one to elevei and back down to one. [his pro- variations in the transformer ratio were being observed.cedure tenids to eliminiate the effect of a steady drift Since it is necessary to wait at least a day after chang-rate of any of the ratio device capacitors. The twenty- ing the oil bath temperature to obtain good tempera-one balances could be made in a period of time rainging ture regulation, the apparent temiperature dependencefrom twenty to thirty minutes, dependirg upon the ex- of the transformer ratio--in particular its phase defect

peablesti the rpepresen terasomraiop-angle-indicated by Table II, is probably at least par-

rabletes Iasdfunctionseotemeratureandsvoltage. Eahop tially due to random variations within the transformer.raimeters as functions of temiperature anid voltage. Each Since the theoretical development of Section II as-number represents the mean of at least five ratio meas- sumes a linear volt-ampere relationship for the com-ureinents. The probable error of each mean is 2 X 109 or ponenits of the ratio device, a method is needed to de-less.less. terminie whether or not the capacitors have voltage

TABLE I coefficients large enouglh to produce an error. SensitivityVARIATION OF a AND , WITH VOLTAGE-24.83°C anid drift problemiis imiake such a determiinlation difficult,

Volt_ but comparisons with capacitors of radically different

Voltage ___________________ a_______________] dconstructioni indicate that any voltage coefficient90 -16X10-9 -12X10-9 which may exist could not produce an error larger than100 +26 -16110 +62 -14 3X10-9 in a or f (50 per cent confidenice interval). We

conclude that with the equipment described above, ten-to-one transformiier ratios may be determined to

TABLE 11 _4X 10-9 with a 50 per cent confidence interval. AVARIATION OF a AND , WITH TEMPERATURE-100 VOLTS imiore extensive inivestigation of voltage coefficienits

l'e_perature l a_i-might reduce this figure.Tlemperature aI d

24.62 +12X10-9 - 4X10-9 ACKNOWLEDGMENT24.83 +26 -1625.42 +34 - 9 The authors wislh to thank Lai H. Lee for his assistance

--_________- in obtaininig the nuniierical results of this paper.

the precision measurement of transformer ratios

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